Fractional Time-Scales Noether’s Theorem for Non-Standard Birkhoffian System

Abstract

In this work, Noether symmetries and conserved quantities of a non-standard Birkhoffian system based on the Caputo $\Delta$ Pfaff–Birkhoff principle on time scales are studied. Firstly, equations of motion for Caputo $\Delta$ non-standard Birkhoffian systems are set up from Caputo $\Delta$ variational principle. Secondly, invariance of Caputo non-standard Pfaff action on time scales is demonstrated, thus giving rise to Noether symmetry criterions which establish Noether’s theorems for the corresponding system. The validity of the methods and results presented in the paper is illustrated by means of examples provided at the end of the article.

1. Introduction

The utilisation of symmetry as a means to identify conserved quantities has emerged as a contemporary approach to studying the conservation of constrained mechanical systems. This methodology has garnered significant attention within the realm of analytical mechanics, a field that is undergoing rapid advancements. The classical symmetry encompasses Noether symmetry [1], Lie symmetry [2], and Mei symmetry [3]. Noether’s theorem has a series of findings in analytical mechanics [4,5]. In 2017, Zhai and Zhang [6] proposed the Noether theorem with delay on time scales and Song [7] studied the Noether theorem of non-standard system. Recently, Anerot and Cresson [8] carried out numerical simulations of the Noether theorems on time scales. Furthermore, Noether-type adiabatic invariants and Herglotz-type Noether theorems were studied in [9,10,11].

Time-scale theory is a mathematical theory proposed by German scholar Hilger [12] in 1988. It is frequently the case that the initiation of a study on one time scale can yield divergent results when conducted on a different time scale. The time-scale theory has the capacity to unify discrete and continuous conditions for study, thus facilitating the revelation of the similarities and differences between discrete and continuous conditions, as well as providing a clearer and more accurate understanding of the essential problems of complex dynamical systems. The superiority of fractional calculus over integer models in the analysis of complex systems is attributable to its unique memory and non-locality. Consequently, fractional differential equations are capable of providing more accurate descriptions of a wider range of practical problems than integer differential equations. A series of findings have been obtained in the research concerning fractional calculus [13,14] and time-scale calculus [15,16,17]. Fractional time-scales calculus has also achieved some results; these include principles and inequalities [18] and dynamical Equations [19]. However, the number of studies examining symmetries on fractional time scales is limited; indeed, only a small number of studies [20,21] address the topic.

The non-standard Lagrangian has been employed for the modelling of non-linear, non-conservative dynamical systems, and has been applied to dissipative systems [22], classical system dynamics [23], and so forth in recent years. Birkhoffian mechanics [24] represents a significant theoretical framework within the broader domain of Hamiltonian mechanics, characterised by its notable degree of generalisation. Consequently, the investigation of the symmetry inherent in non-standard Birkhoffian systems operating on fractional time scales assumes considerable theoretical and practical importance. The symmetry of non-standard systems on fractional time scales has not been studied hitherto. This paper will investigate the Noether symmetries of non-standard Birkhoffian systems based on Caputo delta derivatives, which leads to a more widely applicable method for solving the equations of complex systems.

The following description outlines the arrangement of the text: initially, we list a number of definitions and lemmas that are deemed essential for this paper. Then, in Section 3, we derive fractional non-standard equations of motion with Caputo $\Delta$ derivatives and then degenerate them to the fractional order, time scale order, and the classical case. Subsequently, we derive their Noether identities and conserved quantities in Section 4, which lead to Noether theorems. Then, three illustrative examples are presented to demonstrate the accuracy in Section 5. Finally, we conclude the article and present some perspectives.

2. Preliminaries

In order to facilitate comprehension, the following definition of time scale is listed.

A time scale $\mathbb{T}$ is an arbitrary non-empty closed subset of the real numbers. Thus, the real numbers $\mathbb{R}$, the integers $\mathbb{Z}$, and the natural numbers $\mathbb{N}$ are examples of time scales. For $t\in \mathbb{T}$, we define the forward jump operator $\sigma :\mathbb{T}\to \mathbb{T},\sigma \left(t\right)=inf\left\{s\in \mathbb{T}:s>t\right\}$ and the backward jump operator $\rho :\mathbb{T}\to \mathbb{T},\rho \left(t\right)=sup\left\{s\in \mathbb{T}:s<t\right\}$. Suppose that the function $f:\mathbb{T}\to \mathbb{R}$, if given any $\epsilon >0$, there is a neighbourhood U of t satisfying $\left|f\left(\sigma \left(t\right)\right)-f\left(k\right)-f^{\Delta }\left(t\right)(\sigma \left(t\right)-k)\right|\le \epsilon \left|\sigma \left(t\right)-k\right|$ for all $k\in U$, then $f^{\Delta }\left(t\right)$ is called the delta derivative of f at $t\in {\mathbb{T}}^{\kappa }$. A function $f:\mathbb{T}\to \mathbb{R}$ is called rd-continuous at right-dense points and its left-sided limits exist at left-dense points. The set of all rd-continuous functions in $\mathbb{T}$ is donoted $C_{rd}^0$, and the set of delta differentiable functions in $\mathbb{T}$ with continuous delta derivatives is donated $C_{rd}^{1,\Delta }$.

More introductions of time scales can be found in references [25,26,27]. Next, we focus on the main definitions [28,29,30] and remarks [31,32,33] of the fractional time scales needed for this article.

Definition 1.

Let $\alpha \ge 0$, t,$r\in \mathbb{T}$, defining rd-continuous functions $h_{\alpha }$:

$$h_{\alpha +1}\left(t,r\right)={\int }_r^t{h}_{\alpha }\left(\tau ,r\right)\Delta \tau ,$$

(1)

where $h_0\left(t,r\right)=1$.

Definition 2.

Let $\alpha >0$, $f\left(t\right):\left[a,b\right]\subset \mathbb{R}$, and the left and right Riemann–Liouville Δ integrals are

$${}_{a}I_{\Delta ,t}^{\alpha }f={\int }_a^t{h}_{\alpha -1}\left(t,\sigma \left(\tau \right)\right)f\Delta \tau ,t>a,$$

(2)

$${}_{t}I_{\Delta ,b}^{\alpha }f={\int }_t^b{h}_{\alpha -1}\left(\sigma \left(\tau \right),t\right)f\Delta \tau ,t<b.$$

(3)

Remark 1.

Let $\alpha =0$, we can get ${}_{a}I_{\Delta ,t}^{\alpha }f={}_{t}I_{\Delta ,b}^{\alpha }f=f$. If $\mathbb{T}=\mathbb{R}$, we can obtain

$$\begin{array}{c}{}_{a}I_t^{\alpha }f={\int }_a^t{\displaystyle \frac{{\left(t-\tau \right)}^{\alpha -1}}{\Gamma \left(\alpha \right)}}f\mathrm{d}\tau ,t>a,\end{array}$$

(4)

$$\begin{array}{c}{}_{t}I_b^{\alpha }f={\int }_t^b{\displaystyle \frac{{\left(\tau -t\right)}^{\alpha -1}}{\Gamma \left(\alpha \right)}}f\mathrm{d}\tau ,t<b.\end{array}$$

(5)

Definition 3.

Let $\alpha >0$, for $t\in \mathbb{T}$, and the left and right Caputo Δ derivatives are

$$\begin{array}{c}\hfill {}_a{}^{C}D_{\Delta ,t}^{\alpha }f={}_{a}I_{\Delta ,t}^{1-\alpha }{f}^{\Delta }={}_{a}D_{\Delta ,t}^{\alpha }\left(f-\sum _{k=0}^{m-1}{h}_k(t,a)f^{{\Delta }^k}\left(a\right)\right),\end{array}$$

(6)

$$\begin{array}{c}\hfill {}_t{}^{C}D_{\Delta ,b}^{\alpha }f=-{}_{t}I_{\Delta ,b}^{1-\alpha }{f}^{\Delta }={}_{t}D_{\Delta ,b}^{\alpha }\left(f-\sum _{k=0}^{m-1}{h}_k(b,t)f^{{\Delta }^k}\left(b\right)\right).\end{array}$$

(7)

where ${}_{a}D_{\Delta ,t}^{\alpha }$ is the left Riemann–Liouville Δ derivative and ${}_{t}D_{\Delta ,b}^{\alpha }$ is the right Riemann–Liouville Δ derivative.

Lemma 1.

If $f\left(t\right)$, $g\left(t\right)$ are delta differentiable, then the Leibniz formula for the delta derivative is

$${\left(fg\right)}^{\Delta }=g^{\Delta }{f}^{\sigma }+g{f}^{\Delta }.$$

(8)

If $\phi \left(t\right)$ and $\varphi \left(t\right)$ are delta differentiable, the following formulas represent the integration of fractional time scales by parts:

$${\int }_a^b\varphi {}_a{}^{C}D_{\Delta ,t}^{\alpha }\phi \Delta t={\int }_a^b{\phi }^{\sigma }{}_{t}D_{\Delta ,b}^{\alpha }\varphi \Delta t+{\left.\left[\phi {}_{t}I_{\Delta ,b}^{1-\alpha }\varphi \right]\right|}_a^b,$$

(9)

$${\int }_a^b\varphi {}_t{}^{C}D_{\Delta ,b}^{\alpha }\phi \Delta t={\int }_a^b{\phi }^{\sigma }{}_{a}D_{\Delta ,t}^{\alpha }\varphi \Delta t-{\left.\left[\phi {}_{a}I_{\Delta ,t}^{1-\alpha }\varphi \right]\right|}_a^b.$$

(10)

Lemma 2.

Let $\gamma \left(t\right)\in C_{rd}^0$, $\gamma \left(t\right):\left[a,b\right]\to {\mathbb{R}}^n$, ${\int }_a^b{\gamma }^T{\psi }^{\Delta }\Delta t=0$ holds for $\psi \left(t\right)\in C_{rd}^1$ with $\psi \left(a\right)=\psi \left(b\right)=0$ iff $\gamma \left(t\right)$ is a constant on ${\left[a,b\right]}^m$.

The left Caputo $\Delta$ derivative ${}_a{}^{C}D_{\Delta ,t}^{\alpha }$ is denoted as $C_a$ and the right Riemann–Liouville $\Delta$ derivative ${}_{t}D_{\Delta ,b}^{\alpha }$ is denoted as ${\Delta }_b$ in the following text.

3. Differential Equations of Motion

3.1. Exponential Birkhoffian

Assume that the Pfaff action for the dynamical system based on the fractional exponential Birkhoffian on time scales is

$$A_E={\int }_a^b\left[exp\left(R_{\nu }\left(t,a_{\omega }^{\sigma }\right)C_a{a}_{\nu }-B\left(t,a_{\omega }^{\sigma }\right)\right)\right]\Delta t,$$

(11)

where $a_{\omega }\left(\nu ,\omega =1,2,\dots ,2n\right)$ is the Birkhoffian variable, $R_{\nu }:\mathbb{T}\times {\mathbb{R}}^{2n}\to \mathbb{R}$ are Birkhoffian functions and $B:\mathbb{T}\times {\mathbb{R}}^{2n}\to \mathbb{R}$ is the Birkhoffian, $\alpha \in \left(0,1\right]$.

The Pfaff–Birkhoff principle is

$$\delta A_E=0,$$

(12)

which satisfies commutative conditions

$$\delta C_a{a}_{\nu }=C_a\delta a_{\nu },{\left(\delta a_{\nu }\right)}^{\sigma }=\delta a_{\nu }^{\sigma }$$

(13)

and boundary conditions

$$\delta a_{\nu }\left(a\right)=\delta a_{\nu }\left(b\right)=0.$$

(14)

Substituting Equation (11) into Equation (12), we get

$$\delta A_E={\int }_a^b\left[exp\left(R_{\nu }·C_a{a}_{\nu }-B\right)\left({\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }^{\sigma }}}\delta a_{\omega }^{\sigma }·C_a{a}_{\nu }\right.\right.\left.\left.+R_{\nu }·\delta C_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial a_{\omega }^{\sigma }}}\delta a_{\omega }^{\sigma }\right)\right]\Delta t=0.$$

(15)

According to Equation (9), and then by commutative conditions (13) and boundary conditions (14), we get

$$\begin{array}{cc}\hfill & {\int }_a^b\left[exp\left(R_{\nu }·C_a{a}_{\nu }-B\right)\left(R_{\nu }·\delta C_a{a}_{\nu }\right)\right]\Delta t\hfill \\ \hfill =& {\int }_a^b\left[{\Delta }_b\left(exp\left(R_{\nu }·C_a{a}_{\nu }-B\right)R_{\nu }\right)\delta a_{\omega }^{\sigma }\right]\Delta t\hfill \\ \hfill & +{\left.\left[\delta a_{\omega }·{}_{t}I_{\Delta ,b}^{1-\alpha }\left(exp\left(R_{\nu }·C_a{a}_{\nu }-B\right)R_{\nu }\right)\right]\right|}_a^b\hfill \\ \hfill =& {\int }_a^b\left[{\Delta }_b\left(exp\left(R_{\nu }·C_a{a}_{\nu }-B\right)R_{\nu }\right)\delta a_{\omega }^{\sigma }\right]\Delta t.\hfill \end{array}$$

(16)

Then, we can get

$${\int }_a^b\left[exp\left(R_{\nu }·C_a{a}_{\nu }-B\right)\left({\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }^{\sigma }}}·C_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial a_{\omega }^{\sigma }}}\right)+{\Delta }_b\left(exp\left(R_{\nu }·C_a{a}_{\nu }-B\right)R_{\nu }\right)\right]\delta a_{\omega }^{\sigma }\Delta t=0.$$

(17)

By Lemma 2, taking the delta derivative of Equation (17), we get

$$exp\left(R_{\nu }·C_a{a}_{\nu }-B\right)\left({\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }^{\sigma }}}·C_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial a_{\omega }^{\sigma }}}\right)+{\Delta }_b\left[exp\left(R_{\nu }·C_a{a}_{\nu }-B\right)R_{\nu }\right]=0.$$

(18)

Equation (18) is the fractional differential equation of motion based on the exponential Birkhoffian on time scales. The differential equation is a coupling term on fractional time scales that unifies the constraints and maintains its symmetry for both non-holonomic and complex constrained systems. The non-standard equations of the Birkhoffian system can be employed for the modelling of non-conservative dynamical systems, where the Birkhoffian B can generally be used to represent the energy. The same goes for the next two cases.

Remark 2.

If $\alpha =1$, Equation (18) reduces to the equation of motion based on the exponential Birkhoffian on time scales

$$exp\left(R_{\nu }·a_{\nu }^{\Delta }-B\right)\left({\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }^{\sigma }}}·a_{\nu }^{\Delta }-{\displaystyle \frac{\Delta }{\Delta t}}\left(exp\left(R_{\nu }·a_{\nu }^{\Delta }-B\right)R_{\nu }\right)-{\displaystyle \frac{\partial B}{\partial a_{\omega }^{\sigma }}}\right)=0.$$

(19)

If $\mathbb{T}=\mathbb{R}$, Equation (18) reduces to the fractional equation of motion based on the exponential Birkhoffian

$$exp\left(R_{\nu }{}_a{}^{C}D_t^{\alpha }{a}_{\nu }-B\right)\left({\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }}}{}_a{}^{C}D_t^{\alpha }{a}_{\nu }+{}_{t}D_b^{\alpha }\left(exp\left(R_{\nu }{}_a{}^{C}D_t^{\alpha }{a}_{\nu }-B\right)R_{\nu }\right)-{\displaystyle \frac{\partial B}{\partial a_{\omega }}}\right)=0.$$

(20)

If $\alpha =1$, $\mathbb{T}=\mathbb{R}$, Equation (18) reduces to the classical differential equation of motion based on the exponential Birkhoffian

$$exp\left(R_{\nu }{\dot{a}}_{\nu }-B\right)\left(\left({\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }}}-{\displaystyle \frac{\partial R_{\omega }}{\partial a_{\nu }}}\right){\dot{a}}_{\nu }-{\displaystyle \frac{\partial B}{\partial a_{\omega }}}-{\displaystyle \frac{\partial R_{\omega }}{\partial t}}-{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left(R_{\nu }{\dot{a}}_{\nu }-B\right)R_{\omega }\right)=0$$

(21)

The result is consistent with that in reference [24].

3.2. Logarithmic Birkhoffian

Assume that the Pfaff action for the dynamical system based on the fractional logarithmic Birkhoffian on time scales is

$$A_L={\int }_a^b\left[lo{g}_b\left(R_{\nu }\left(t,a_{\omega }^{\sigma }\right)C_a{a}_{\nu }-B\left(t,a_{\omega }^{\sigma }\right)\right)\right]\Delta t,$$

(22)

where $a_{\omega }\left(\nu ,\omega =1,2,\dots ,2n\right)$ is the Birkhoffian variable, $R_{\nu }:\mathbb{T}\times {\mathbb{R}}^{2n}\to \mathbb{R}$ are Birkhoffian functions, and $B:\mathbb{T}\times {\mathbb{R}}^{2n}\to \mathbb{R}$ is the Birkhoffian, $\alpha \in \left(0,1\right]$.

The Pfaff–Birkhoff principle is

$$\delta A_L=0,$$

(23)

which satisfies commutative conditions

$$\delta C_a{a}_{\nu }=C_a\delta a_{\nu },{\left(\delta a_{\nu }\right)}^{\sigma }=\delta a_{\nu }^{\sigma }$$

(24)

and boundary conditions

$$\delta a_{\nu }\left(a\right)=\delta a_{\nu }\left(b\right)=0.$$

(25)

Substituting Equation (22) into Equation (23), we have

$$\delta A_L={\int }_a^b\left[{\displaystyle \frac{1}{\left(R_{\nu }{C}_a{a}_{\nu }-B\right)lnb}}\right.\left({\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }^{\sigma }}}\delta a_{\omega }^{\sigma }·C_a{a}_{\nu }\right.\left.\left.+R_{\nu }·\delta C_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial a_{\omega }^{\sigma }}}\delta a_{\omega }^{\sigma }\right)\right]\Delta t=0.$$

(26)

According to Equation (9), and then by commutative conditions (24) and boundary conditions (25), we get

$$\begin{array}{cc}\hfill & {\int }_a^b\left[{\displaystyle \frac{1}{\left(R_{\nu }{C}_a{a}_{\nu }-B\right)lnb}}\left(R_{\nu }·\delta C_a{a}_{\nu }\right)\right]\Delta t\hfill \\ \hfill =& {\int }_a^b\left[{\Delta }_b\left({\displaystyle \frac{R_{\nu }}{\left(R_{\nu }{C}_a{a}_{\nu }-B\right)lnb}}\right)\delta a_{\omega }^{\sigma }\right]\Delta t\hfill \\ \hfill & +{\left.\left[\delta a_{\omega }·{}_{t}I_{\Delta ,b}^{1-\alpha }\left({\displaystyle \frac{R_{\nu }}{\left(R_{\nu }{C}_a{a}_{\nu }-B\right)lnb}}\right)\right]\right|}_a^b\hfill \\ \hfill =& {\int }_a^b\left[{\Delta }_b\left({\displaystyle \frac{R_{\nu }}{\left(R_{\nu }{C}_a{a}_{\nu }-B\right)lnb}}\right)\delta a_{\omega }^{\sigma }\right]\Delta t.\hfill \end{array}$$

(27)

Then, we can get

$${\int }_a^b\left[{\displaystyle \frac{1}{\left(R_{\nu }{C}_a{a}_{\nu }-B\right)lnb}}\left({\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }^{\sigma }}}·C_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial a_{\omega }^{\sigma }}}\right)\delta a_{\omega }^{\sigma }\right.\left.+{\Delta }_b\left({\displaystyle \frac{R_{\nu }}{\left(R_{\nu }{C}_a{a}_{\nu }-B\right)lnb}}\right)\delta a_{\omega }^{\sigma }\right]\Delta t=0.$$

(28)

By Lemma 2, taking the delta derivative of Equation (28), we get

$${\displaystyle \frac{1}{R_{\nu }{C}_a{a}_{\nu }-B}}\left({\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }^{\sigma }}}{C}_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial a_{\omega }^{\sigma }}}\right)+{\Delta }_b\left({\displaystyle \frac{R_{\nu }}{R_{\nu }{C}_a{a}_{\nu }-B}}\right)=0.$$

(29)

Equation (29) is the fractional differential equation of motion based on the logarithmic Birkhoffian on time scales.

Remark 3.

If $\alpha =1$, Equation (29) reduces to the equation of motion based on the logarithmic Birkhoffian on time scales

$${\displaystyle \frac{1}{R_{\nu }{a}_{\nu }^{\Delta }-B}}\left({\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }^{\sigma }}}·a_{\nu }^{\Delta }-{\displaystyle \frac{\partial B}{\partial a_{\omega }^{\sigma }}}\right)-{\displaystyle \frac{\Delta }{\Delta t}}\left({\displaystyle \frac{R_{\omega }}{R_{\omega }{a}_{\omega }^{\Delta }-B}}\right)=0.$$

(30)

If $\mathbb{T}=\mathbb{R}$, Equation (29) reduces to the fractional equation of motion based on the logarithmic Birkhoffian

$${\displaystyle \frac{1}{R_{\nu }{}_a{}^{C}D_t^{\alpha }{a}_{\nu }-B}}\left({\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }}}{}_a{}^{C}D_t^{\alpha }{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial a_{\omega }}}\right)+{}_{t}D_b^{\alpha }\left({\displaystyle \frac{R_{\omega }}{R_{\omega }{}_a{}^{C}D_t^{\alpha }{a}_{\omega }-B}}\right)=0.$$

(31)

If $\alpha =1$, $\mathbb{T}=\mathbb{R}$, Equation (29) reduces to the classical differential equation of motion based on the logarithmic Birkhoffian

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{R_{\nu }{\dot{a}}_{\nu }-B}}\left(\left({\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }}}-{\displaystyle \frac{\partial R_{\omega }}{\partial a_{\nu }}}\right){\dot{a}}_{\nu }-{\displaystyle \frac{\partial B}{\partial a_{\omega }}}-{\displaystyle \frac{\partial R_{\omega }}{\partial t}}\right.\left.-{\displaystyle \frac{1}{\left(R_{\nu }{\dot{a}}_{\nu }-B\right)}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left(R_{\nu }{\dot{a}}_{\nu }-B\right)R_{\omega }\right)=0.\end{array}$$

(32)

The result is consistent with that in reference [24].

3.3. Power Birkhoffian

Assume that the Pfaff action for the dynamical system based on the fractional power Birkhoffian on time scales is

$$A_P={\int }_a^b\left[{\left(R_{\nu }\left(t,a_{\omega }^{\sigma }\right)C_a{a}_{\nu }-B\left(t,a_{\omega }^{\sigma }\right)\right)}^{1+\gamma }\right]\Delta t,$$

(33)

where $a_{\omega }\left(\nu ,\omega =1,2,\dots ,2n\right)$ is the Birkhoffian variable, $R_{\nu }:\mathbb{T}\times {\mathbb{R}}^{2n}\to \mathbb{R}$ are Birkhoffian functions, and $B:\mathbb{T}\times {\mathbb{R}}^{2n}\to \mathbb{R}$ is the Birkhoffian, $\alpha \in \left(0,1\right]$.

The Pfaff–Birkhoff principle is

$$\delta A_P=0,$$

(34)

which satisfies commutative conditions

$$\delta C_a{a}_{\nu }=C_a\delta a_{\nu },{\left(\delta a_{\nu }\right)}^{\sigma }=\delta a_{\nu }^{\sigma }$$

(35)

and boundary conditions

$$\delta a_{\nu }\left(a\right)=\delta a_{\nu }\left(b\right)=0.$$

(36)

Substituting Equation (33) into Equation (34), we have

$$\delta A_P={\int }_a^b\left[\left(1+\gamma \right){\left(R_{\nu }{C}_a{a}_{\nu }-B\right)}^{\gamma }\left({\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }^{\sigma }}}\delta a_{\omega }^{\sigma }·C_a{a}_{\nu }\right.\right.\left.\left.+R_{\nu }·\delta C_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial a_{\omega }^{\sigma }}}\delta a_{\omega }^{\sigma }\right)\right]\Delta t=0.$$

(37)

According to Equation (9), and then by commutative conditions (35) and boundary conditions (36), we get

$$\begin{array}{cc}\hfill & {\int }_a^b\left[\left(1+\gamma \right){\left(R_{\nu }{C}_a{a}_{\nu }-B\right)}^{\gamma }\left(R_{\nu }·\delta C_a{a}_{\nu }\right)\right]\Delta t\hfill \\ \hfill =& {\int }_a^b\left[{\Delta }_b\left(\left(1+\gamma \right){\left(R_{\nu }{C}_a{a}_{\nu }-B\right)}^{\gamma }{R}_{\nu }\right)\delta a_{\omega }^{\sigma }\right]\Delta t\hfill \\ \hfill & +{\left.\left[\delta a_{\omega }·{}_{t}I_{\Delta ,b}^{1-\alpha }\left(\left(1+\gamma \right){\left(R_{\nu }{C}_a{a}_{\nu }-B\right)}^{\gamma }{R}_{\nu }\right)\right]\right|}_a^b\hfill \\ \hfill =& {\int }_a^b\left[{\Delta }_b\left(\left(1+\gamma \right){\left(R_{\nu }{C}_a{a}_{\nu }-B\right)}^{\gamma }{R}_{\nu }\right)\delta a_{\omega }^{\sigma }\right]\Delta t.\hfill \end{array}$$

(38)

Then, we can get

$${\int }_a^b\left[\left(1+\gamma \right){\left(R_{\nu }{C}_a{a}_{\nu }-B\right)}^{\gamma }\left({\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }^{\sigma }}}·C_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial a_{\omega }^{\sigma }}}\right)\right.\left.+{\Delta }_b\left(\left(1+\gamma \right){\left(R_{\nu }{C}_a{a}_{\nu }-B\right)}^{\gamma }{R}_{\nu }\right)\right]\delta a_{\omega }^{\sigma }\Delta t=0.$$

(39)

By Lemma 2, taking the delta derivative of Equation (39), we have

$${\left(R_{\nu }{C}_a{a}_{\nu }-B\right)}^{\gamma }\left({\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }^{\sigma }}}·C_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial a_{\omega }^{\sigma }}}\right)+{\Delta }_b\left({\left(R_{\nu }{C}_a{a}_{\nu }-B\right)}^{\gamma }{R}_{\nu }\right)=0.$$

(40)

Equation (40) is the fractional differential equation of motion based on the power Birkhoffian on time scales.

Remark 4.

If $\alpha =1$, Equation (40) reduces to the equation of motion based on the power Birkhoffian on time scales

$${\left(R_{\nu }{a}_{\nu }^{\Delta }-B\right)}^{\gamma }\left({\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }^{\sigma }}}·a_{\nu }^{\Delta }-{\displaystyle \frac{\partial B}{\partial a_{\omega }^{\sigma }}}\right)-{\displaystyle \frac{\Delta }{\Delta t}}\left({\left(R_{\omega }{a}_{\omega }^{\Delta }-B\right)}^{\gamma }{R}_{\omega }\right)=0.$$

(41)

If $\mathbb{T}=\mathbb{R}$, Equation (40) reduces to the fractional equation of motion based on the power Birkhoffian

$${\left(R_{\nu }{}_a{}^{C}D_t^{\alpha }{a}_{\nu }-B\right)}^{\gamma }\left({\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }}}{}_a{}^{C}D_t^{\alpha }{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial a_{\omega }}}\right)+{}_{t}D_b^{\alpha }\left({\left(R_{\omega }{}_a{}^{C}D_t^{\alpha }{a}_{\omega }-B\right)}^{\gamma }{R}_{\omega }\right)=0.$$

(42)

If $\alpha =1$, $\mathbb{T}=\mathbb{R}$, Equation (40) reduces to the classical differential equation of motion based on the power Birkhoffian

$${\left(R_{\nu }{\dot{a}}_{\nu }-B\right)}^{\gamma -1}\left[\left(R_{\nu }{\dot{a}}_{\nu }-B\right)\left({\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }}}-{\displaystyle \frac{\partial R_{\omega }}{\partial a_{\nu }}}\right){\dot{a}}_{\nu }\right.\left.-{\displaystyle \frac{\partial B}{\partial a_{\omega }}}-{\displaystyle \frac{\partial R_{\omega }}{\partial t}}-\gamma {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left(R_{\nu }{\dot{a}}_{\nu }-B\right)R_{\omega }\right]=0.$$

(43)

The result is consistent with that in reference [24].

The introduction of fractional theory allows the equations to be degraded to integer order models. The time-scale theory enables equations to degenerate into discrete case, where it is constructed in a way that avoids the numerical dissipation caused by artificial discrete motions, and inherits realistically the geometrical properties of the solution of a continuous system, such as the structure-preserving system.

4. Noether’s Theorem

We introduce the infinitesimal transformation

$$\begin{array}{c}\overline{t}=T_{\epsilon }=t+\epsilon {\xi }_0\left(t,a_{\nu }\right)+o\left(\epsilon \right),\\ {\overline{a}}_{\omega }\left(\overline{t}\right)=A_{\epsilon }^{\omega }=a_{\omega }\left(t\right)+\epsilon {\xi }_{\omega }\left(t,a_{\nu }\right)+o\left(\epsilon \right),\end{array}$$

(44)

where $\epsilon$ is the infinitesimal parameter and ${\xi }_0$, ${\xi }_{\omega }$ are infinitesimal generators of the transformation. Define the mapping $t\mapsto \rho \left(t\right)=t+\epsilon {\xi }_0$ as a new time scale $\overline{\mathbb{T}}$ and as a strictly increasing $C_{rd}^{1,\Delta }$ function whose forward jump operator is $\overline{\sigma }$, and whose corresponding $\Delta$ derivative is $\overline{\Delta }$.

4.1. Exponential Birkhoffian System

Definition 4.

For arbitrary closed interval $\left[t_a,t_b\right]\subseteq \left[a,b\right]$, if

$$\begin{array}{cc}\hfill & {\int }_{t_a}^{t_b}\left[exp\left(R_{\nu }\left(t,a_{\omega }^{\sigma }\right)C_a{a}_{\nu }\left(t\right)-B\left(t,a_{\omega }^{\sigma }\right)\right)\right]\Delta t\hfill \\ \hfill =& {\int }_{\rho \left(t_a\right)}^{\rho \left(t_b\right)}\left[exp\left(R_{\nu }\left(\overline{t},{\overline{a}}_{\omega }^{\overline{\sigma }}\left(\overline{t}\right)\right){\overline{C}}_{\rho \left(a\right)}{\overline{a}}_{\nu }\left(\overline{t}\right)-B\left(\overline{t},{\overline{a}}_{\omega }^{\overline{\sigma }}\right)\right)\right]\overline{\Delta }\overline{t},\hfill \end{array}$$

(45)

then it is claimed that action Equation (11) is invariant under the general infinitesimal transformation.

If Equation (11) is invariant under the transformation, then the generators ${\xi }_0$, ${\xi }_{\omega }$ satisfy the Noether identity

$$\begin{array}{cc}\hfill & exp\left(R_{\nu }{C}_a{a}_{\nu }-B\right)\left[{\xi }_0^{\Delta }+\left({\displaystyle \frac{\partial R_{\nu }}{\partial t}}{\xi }_0+{\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }^{\sigma }}}{\xi }_{\omega }^{\sigma }\right)C_a{a}_{\nu }\right.\hfill \\ \hfill & \left.+R_{\nu }{C}_a{\xi }_{\nu }-\alpha R_{\nu }{C}_a{a}_{\nu }{\xi }_0^{\Delta }-{\displaystyle \frac{\partial B}{\partial t}}{\xi }_0-{\displaystyle \frac{\partial B}{\partial a_{\omega }^{\sigma }}}{\xi }_{\omega }^{\sigma }\right]=0.\hfill \end{array}$$

(46)

Proof. 

Expanding Equation (45), we have

$$\begin{array}{cc}\hfill & {\int }_{t_a}^{t_b}\left[exp\left(R_{\nu }\left(t,a_{\omega }^{\sigma }\right)C_a{a}_{\nu }\left(t\right)-B\left(t,a_{\omega }^{\sigma }\right)\right)\right]\Delta t\hfill \\ \hfill =& {\int }_{\rho \left(t_a\right)}^{\rho \left(t_b\right)}\left[exp\left(R_{\nu }\left(\overline{t},{\overline{a}}_{\omega }^{\overline{\sigma }}\left(\overline{t}\right)\right){\overline{C}}_{\rho \left(a\right)}{\overline{a}}_{\nu }\left(\overline{t}\right)-B\left(\overline{t},{\overline{a}}_{\omega }^{\overline{\sigma }}\right)\right)\right]\overline{\Delta }\overline{t}\hfill \\ \hfill =& {\int }_{t_a}^{t_b}\left[exp\left(R_{\nu }\left(\rho \left(t\right),\left({\overline{a}}_{\omega }\circ \overline{\sigma }\circ \rho \right)\left(t\right)\right)\right.\right.{\displaystyle \frac{C_a\left({\overline{a}}_{\nu }\circ \rho \right)\left(\overline{t}\right)}{{\left({\rho }^{\Delta }\left(t\right)\right)}^{\alpha }}}\left.\left.-B\left(\rho \left(t\right),\left({\overline{a}}_{\omega }\circ \overline{\sigma }\circ \rho \right)\left(t\right)\right)\right)\right]{\rho }^{\Delta }\left(t\right)\Delta t\hfill \\ \hfill =& {\int }_{t_a}^{t_b}\left[exp\left(R_{\nu }\left(T_{\epsilon },{\left(A_{\epsilon }^{\omega }\right)}^{\sigma }\right){\displaystyle \frac{C_a{A}_{\epsilon }^{\omega }}{{\left(T_{\epsilon }^{\Delta }\right)}^{\alpha }}}-B\left(T_{\epsilon },{\left(A_{\epsilon }^{\omega }\right)}^{\sigma }\right)\right)\right]T_{\epsilon }^{\Delta }\Delta t.\hfill \end{array}$$

(47)

Since $\left[t_a,t_b\right]\subseteq \left[a,b\right]$, Equation (47) is equal to

$$\begin{array}{cc}\hfill & exp\left(R_{\nu }\left(t,a_{\omega }^{\sigma }\right)C_a{a}_{\nu }\left(t\right)-B\left(t,a_{\omega }^{\sigma }\right)\right)\hfill \\ \hfill =& \left[exp\left(R_{\nu }\left(T_{\epsilon },{\left(A_{\epsilon }^{\omega }\right)}^{\sigma }\right){\displaystyle \frac{C_a{A}_{\epsilon }^{\omega }}{{\left(T_{\epsilon }^{\Delta }\right)}^{\alpha }}}-B\left(T_{\epsilon },{\left(A_{\epsilon }^{\omega }\right)}^{\sigma }\right)\right)\right]T_{\epsilon }^{\Delta }.\hfill \end{array}$$

(48)

Taking the derivative of Equation (48) with respect to $\epsilon$ and put $\epsilon =0$, then we obtain Equation (46). □

Theorem 1.

If Equation (11) is invariant under Definition 4, then

$$\begin{array}{cc}\hfill I_E=& exp\left(\Im \right)\left[1-\alpha R_{\nu }{C}_a{a}_{\nu }+\mu \left({\displaystyle \frac{\partial R_{\nu }}{\partial t}}{C}_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial t}}\right)\right]{\xi }_0\hfill \\ \hfill & +{\int }_a^{t}exp\left(\Im \right)\left({\displaystyle \frac{\partial R_{\nu }}{\partial \tau }}{C}_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial \tau }}\right){\xi }_0^{\sigma }\Delta \tau \hfill \\ \hfill & -{\int }_a^t{\left\{exp\left(\Im \right)\left[1-\alpha R_{\nu }{C}_a{a}_{\nu }+\mu \left({\displaystyle \frac{\partial R_{\nu }}{\partial \tau }}{C}_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial \tau }}\right)\right]\right\}}^{\Delta }{\xi }_0^{\sigma }\Delta \tau \hfill \\ \hfill & -{\int }_a^t\left\{{\Delta }_b\left[exp\left(\Im \right)R_{\nu }\right]{\xi }_{\omega }^{\sigma }-exp\left(\Im \right)R_{\nu }{C}_a{\xi }_{\nu }\right\}\Delta \tau \hfill \end{array}$$

(49)

is a Noether conserved quantity, where $\Im =R_{\nu }{C}_a{a}_{\nu }-B$.

Proof. 

Taking the delta derivative of Equation (49), making use of Equations (18) and (46), we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{\Delta }{\Delta t}}{I}_E=& {\left\{exp\left(\Im \right)\left[1-\alpha R_{\nu }{C}_a{a}_{\nu }+\mu \left({\displaystyle \frac{\partial R_{\nu }}{\partial t}}{C}_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial t}}\right)\right]\right\}}^{\Delta }{\xi }_0^{\sigma }\hfill \\ \hfill & +exp\left(\Im \right)\left[1-\alpha R_{\nu }{C}_a{a}_{\nu }+\mu \left({\displaystyle \frac{\partial R_{\nu }}{\partial t}}{C}_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial t}}\right)\right]{\xi }_0^{\Delta }\hfill \\ \hfill & +exp\left(\Im \right)\left({\displaystyle \frac{\partial R_{\nu }}{\partial t}}{C}_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial t}}\right){\xi }_0^{\sigma }\hfill \\ \hfill & -{\left\{exp\left(\Im \right)\left[1-\alpha R_{\nu }{C}_a{a}_{\nu }+\mu \left({\displaystyle \frac{\partial R_{\nu }}{\partial t}}{C}_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial t}}\right)\right]\right\}}^{\Delta }{\xi }_0^{\sigma }\hfill \\ \hfill & -{\Delta }_b\left[exp\left(\Im \right)\right]{\xi }_{\omega }^{\sigma }+exp\left(\Im \right)R_{\nu }{C}_a{\xi }_{\nu }\hfill \\ \hfill =& \left\{exp\left(\Im \right)\left({\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }^{\sigma }}}·C_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial a_{\omega }^{\sigma }}}\right)+{\Delta }_b\left[exp\left(\Im \right)R_{\nu }\right]\right\}{\xi }_{\nu }^{\sigma }\hfill \\ \hfill =& 0.\hfill \end{array}$$

(50)

Hence, Equation (49) is a Noether conserved quantity of the exponential Birkhoffian system. □

Remark 5.

If $\mathbb{T}=\mathbb{R}$, Noether identity (46) reduces to

$$\begin{array}{cc}\hfill & exp\left(R_{\nu }{}_a{}^{C}D_t^{\alpha }{a}_{\nu }-B\right)\left[{\dot{\xi }}_0+\left({\displaystyle \frac{\partial R_{\nu }}{\partial t}}{\xi }_0+{\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }}}{\xi }_{\omega }\right){}_a{}^{C}D_t^{\alpha }{a}_{\nu }\right.\hfill \\ \hfill & \left.+R_{\nu }{}_a{}^{C}D_t^{\alpha }{\xi }_{\nu }-\alpha R_{\nu }{}_a{}^{C}D_t^{\alpha }{a}_{\nu }{\dot{\xi }}_0-{\displaystyle \frac{\partial B}{\partial t}}{\xi }_0-{\displaystyle \frac{\partial B}{\partial a_{\omega }}}{\xi }_{\omega }\right]=0\hfill \end{array}$$

(51)

and the Noether conserved quantity (49) reduces to

$$\begin{array}{cc}\hfill I_E=& exp\left(R_{\nu }{}_a{}^{C}D_t^{\alpha }{a}_{\nu }-B\right)\left(1-\alpha R_{\nu }{}_a{}^{C}D_t^{\alpha }{a}_{\nu }\right){\xi }_0\hfill \\ \hfill & +{\int }_a^{t}exp\left(R_{\nu }{}_a{}^{C}D_{\tau }^{\alpha }{a}_{\nu }-B\right)\left({\displaystyle \frac{\partial R_{\nu }}{\partial \tau }}{}_a{}^{C}D_{\tau }^{\alpha }{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial \tau }}\right){\xi }_0\mathrm{d}\tau \hfill \\ \hfill & -{\int }_a^t{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left[exp\left(R_{\nu }{}_a{}^{C}D_{\tau }^{\alpha }{a}_{\nu }-B\right)\left(1-\alpha R_{\nu }{}_a{}^{C}D_{\tau }^{\alpha }{a}_{\nu }\right)\right]{\xi }_0\mathrm{d}\tau \hfill \\ \hfill & -{\int }_a^t\left\{{}_{\tau }D_b^{\alpha }\left[exp\left(R_{\nu }{}_a{}^{C}D_{\tau }^{\alpha }{a}_{\nu }-B\right)\right]{\xi }_{\omega }\right.\left.-exp\left(R_{\nu }{}_a{}^{C}D_{\tau }^{\alpha }{a}_{\nu }-B\right)R_{\nu }{}_a{}^{C}D_{\tau }^{\alpha }{\xi }_{\nu }\right\}\mathrm{d}\tau .\hfill \end{array}$$

(52)

If $\alpha =1$, Noether identity (46) reduces to

$$exp\left(R_{\nu }{a}_{\nu }^{\Delta }-B\right)\left[{\xi }_0^{\Delta }+\left({\displaystyle \frac{\partial R_{\nu }}{\partial t}}{\xi }_0+{\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }^{\sigma }}}{\xi }_{\omega }^{\sigma }\right)a_{\nu }^{\Delta }\right.\left.+R_{\nu }{\xi }_{\nu }^{\Delta }-R_{\nu }{a}_{\nu }^{\Delta }{\xi }_0^{\Delta }-{\displaystyle \frac{\partial B}{\partial t}}{\xi }_0-{\displaystyle \frac{\partial B}{\partial a_{\omega }^{\sigma }}}{\xi }_{\omega }^{\sigma }\right]=0$$

(53)

and the Noether conserved quantity (49) reduces to

$$\begin{array}{cc}\hfill I_E=& exp\left(R_{\nu }{a}_{\nu }^{\Delta }-B\right)\left[1-R_{\nu }{a}_{\nu }^{\Delta }+\mu \left({\displaystyle \frac{\partial R_{\nu }}{\partial t}}{a}_{\nu }^{\Delta }-{\displaystyle \frac{\partial B}{\partial t}}\right)\right]{\xi }_0\hfill \\ \hfill & +{\int }_a^{t}exp\left(R_{\nu }{a}_{\nu }^{\Delta }-B\right)\left({\displaystyle \frac{\partial R_{\nu }}{\partial \tau }}{a}_{\nu }^{\Delta }-{\displaystyle \frac{\partial B}{\partial \tau }}\right){\xi }_0^{\sigma }\Delta \tau \hfill \\ \hfill & -{\int }_a^t\left\{exp\left(R_{\nu }{a}_{\nu }^{\Delta }-B\right)\right.{\left.\left[1-R_{\nu }{a}_{\nu }^{\Delta }+\mu \left({\displaystyle \frac{\partial R_{\nu }}{\partial \tau }}{a}_{\nu }^{\Delta }-{\displaystyle \frac{\partial B}{\partial \tau }}\right)\right]\right\}}^{\Delta }{\xi }_0^{\sigma }\Delta \tau \hfill \\ \hfill & +{\int }_a^t\left\{{\displaystyle \frac{\Delta }{\Delta \tau }}\left[exp\left(R_{\nu }{a}_{\nu }^{\Delta }-B\right)\right]{\xi }_{\omega }^{\sigma }+exp\left(R_{\nu }{a}_{\nu }^{\Delta }-B\right)R_{\nu }{\xi }_{\nu }^{\Delta }\right\}\Delta \tau .\hfill \end{array}$$

(54)

4.2. Logarithmic Birkhoffian System

Definition 5.

For any $\left[t_a,t_b\right]\subseteq \left[a,b\right]$, if

$$\begin{array}{cc}\hfill & {\int }_{t_a}^{t_b}\left[{log}_b\left(R_{\nu }\left(t,a_{\omega }^{\sigma }\right)C_a{a}_{\nu }-B\left(t,a_{\omega }^{\sigma }\right)\right)\right]\Delta t\hfill \\ \hfill =& {\int }_{\rho \left(t_a\right)}^{\rho \left(t_b\right)}\left[{log}_b\left(R_{\nu }\left(\overline{t},{\overline{a}}_{\omega }^{\overline{\sigma }}\left(\overline{t}\right)\right){\overline{C}}_{\rho \left(a\right)}{\overline{a}}_{\nu }\left(\overline{t}\right)-B\left(\overline{t},{\overline{a}}_{\omega }^{\overline{\sigma }}\right)\right)\right]\overline{\Delta }\overline{t},\hfill \end{array}$$

(55)

then it is claimed that action Equation (22) is invariant under the general infinitesimal transformation.

If Equation (22) is invariant under the transformation, then the generators ${\xi }_0$, ${\xi }_{\omega }$ satisfy the Noether identity

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{R_{\nu }{C}_a-B}}\left[\left({\displaystyle \frac{\partial R_{\nu }}{\partial t}}{\xi }_0+{\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }^{\sigma }}}{\xi }_{\omega }^{\sigma }\right)C_a{a}_{\nu }+R_{\nu }{C}_a{\xi }_{\nu }\right.\left.-\alpha R_{\nu }{C}_a{a}_{\nu }{\xi }_0^{\Delta }-{\displaystyle \frac{\partial B}{\partial t}}{\xi }_0-{\displaystyle \frac{\partial B}{\partial a_{\omega }^{\sigma }}}{\xi }_{\omega }^{\sigma }\right]\hfill \\ \hfill & +ln\left(R_{\nu }{C}_a-B\right){\xi }_0^{\Delta }=0.\hfill \end{array}$$

(56)

Proof. 

Expanding Equation (55), we have

$$\begin{array}{cc}\hfill & {\int }_{t_a}^{t_b}\left[{log}_b\left(R_{\nu }\left(t,a_{\omega }^{\sigma }\right)C_a{a}_{\nu }-B\left(t,a_{\omega }^{\sigma }\right)\right)\right]\Delta t\hfill \\ \hfill =& {\int }_{\rho \left(t_a\right)}^{\rho \left(t_b\right)}\left[{log}_b\left(R_{\nu }\left(\overline{t},{\overline{a}}_{\omega }^{\overline{\sigma }}\left(\overline{t}\right)\right){\overline{C}}_a{\overline{a}}_{\nu }\left(\overline{t}\right)-B\left(\overline{t},{\overline{a}}_{\omega }^{\overline{\sigma }}\right)\right)\right]\overline{\Delta }\overline{t}\hfill \\ \hfill =& {\int }_{t_a}^{t_b}\left[{log}_b\left(R_{\nu }\left(T_{\epsilon },{\left(A_{\epsilon }^{\omega }\right)}^{\sigma }\right){\displaystyle \frac{C_a{A}_{\epsilon }^{\omega }}{{\left(T_{\epsilon }^{\Delta }\right)}^{\alpha }}}-B\left(T_{\epsilon },{\left(A_{\epsilon }^{\omega }\right)}^{\sigma }\right)\right)\right]T_{\epsilon }^{\Delta }\Delta t.\hfill \end{array}$$

(57)

Since $\left[t_a,t_b\right]\subseteq \left[a,b\right]$, Equation (57) is equal to

$$\begin{array}{cc}\hfill & {log}_b\left(R_{\nu }\left(t,a_{\omega }^{\sigma }\right)C_a{a}_{\nu }\left(t\right)-B\left(t,a_{\omega }^{\sigma }\right)\right)\hfill \\ \hfill =& \left[{log}_b\left(R_{\nu }\left(T_{\epsilon },{\left(A_{\epsilon }^{\omega }\right)}^{\sigma }\right){\displaystyle \frac{C_a{A}_{\epsilon }^{\omega }}{{\left(T_{\epsilon }^{\Delta }\right)}^{\alpha }}}-B\left(T_{\epsilon },{\left(A_{\epsilon }^{\omega }\right)}^{\sigma }\right)\right)\right]T_{\epsilon }^{\Delta }.\hfill \end{array}$$

(58)

Taking the derivative of Equation (58) with respect to $\epsilon$ and put $\epsilon =0$, then we obtain Equation (56). □

Theorem 2.

If Equation (22) is invariant under Definition 5, then

$$\begin{array}{cc}\hfill I_L=& \left\{ln\left(\Im \right)-{\displaystyle \frac{1}{\Im }}\left[\alpha R_{\nu }{C}_a{a}_{\nu }+\mu \left({\displaystyle \frac{\partial R_{\nu }}{\partial t}}{C}_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial t}}\right)\right]\right\}{\xi }_0\hfill \\ \hfill & +{\int }_a^t{\displaystyle \frac{1}{\Im }}\left({\displaystyle \frac{\partial R_{\nu }}{\partial \tau }}{C}_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial \tau }}\right){\xi }_0^{\sigma }\Delta \tau \hfill \\ \hfill & -{\int }_a^t{\left\{ln\Im -{\displaystyle \frac{1}{\Im }}\left[\alpha R_{\nu }{C}_a{a}_{\nu }+\mu \left({\displaystyle \frac{\partial R_{\nu }}{\partial \tau }}{C}_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial \tau }}\right)\right]\right\}}^{\Delta }{\xi }_0^{\sigma }\Delta \tau \hfill \\ \hfill & -{\int }_a^t\left\{{\Delta }_b\left({\displaystyle \frac{R_{\nu }}{\Im }}\right){\xi }_{\omega }^{\sigma }-{\displaystyle \frac{1}{\Im }}{R}_{\nu }{C}_a{\xi }_{\nu }\right\}\Delta \tau \hfill \end{array}$$

(59)

is a Noether conserved quantity.

Proof. 

Taking the delta derivative of Equation (59), making use of Equations (29) and (56), we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{\Delta }{\Delta t}}{I}_L=& {\left\{ln\left(\Im \right)-{\displaystyle \frac{1}{\Im }}\left[\alpha R_{\nu }{C}_a{a}_{\nu }+\mu \left({\displaystyle \frac{\partial R_{\nu }}{\partial t}}{C}_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial t}}\right)\right]\right\}}^{\Delta }{\xi }_0^{\sigma }\hfill \\ \hfill & +\left\{ln\left(\Im \right)-{\displaystyle \frac{1}{\Im }}\left[\alpha R_{\nu }{C}_a{a}_{\nu }+\mu \left({\displaystyle \frac{\partial R_{\nu }}{\partial t}}{C}_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial t}}\right)\right]\right\}{\xi }_0^{\Delta }\hfill \\ \hfill & +{\displaystyle \frac{1}{\Im }}\left({\displaystyle \frac{\partial R_{\nu }}{\partial \tau }}{C}_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial \tau }}\right){\xi }_0^{\sigma }\hfill \\ \hfill & -{\left\{ln\left(\Im \right)-{\displaystyle \frac{1}{\Im }}\left[\alpha R_{\nu }{C}_a{a}_{\nu }+\mu \left({\displaystyle \frac{\partial R_{\nu }}{\partial t}}{C}_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial t}}\right)\right]\right\}}^{\Delta }{\xi }_0^{\sigma }\hfill \\ \hfill & -\left\{{\Delta }_b\left({\displaystyle \frac{R_{\nu }}{\Im }}\right){\xi }_{\omega }^{\sigma }-{\displaystyle \frac{1}{\Im }}{R}_{\nu }{C}_a{\xi }_{\nu }\right\}\hfill \\ \hfill =& \left\{{\displaystyle \frac{1}{\Im }}\left({\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }^{\sigma }}}·C_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial a_{\omega }^{\sigma }}}\right)+{\Delta }_b\left({\displaystyle \frac{R_{\nu }}{\Im }}\right)\right\}{\xi }_{\omega }^{\sigma }\hfill \\ \hfill =& 0.\hfill \end{array}$$

(60)

Hence, Equation (59) is a Noether conserved quantity of the exponential Birkhoffian system. □

Remark 6.

If $\mathbb{T}=\mathbb{R}$, Noether identity (56) reduces to

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{R_{\nu }{}_a{}^{C}D_t^{\alpha }{a}_{\nu }-B}}\left[\left({\displaystyle \frac{\partial R_{\nu }}{\partial t}}{\xi }_0+{\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }}}{\xi }_{\omega }\right){}_a{}^{C}D_t^{\alpha }{a}_{\nu }+R_{\nu }{}_a{}^{C}D_t^{\alpha }{\xi }_{\nu }\right.\left.-\alpha R_{\nu }{}_a{}^{C}D_t^{\alpha }{a}_{\nu }{\dot{\xi }}_0-{\displaystyle \frac{\partial B}{\partial t}}{\xi }_0-{\displaystyle \frac{\partial B}{\partial a_{\omega }}}{\xi }_{\omega }\right]\hfill \\ \hfill & +ln\left(R_{\nu }{}_a{}^{C}D_t^{\alpha }{a}_{\nu }-B\right){\dot{\xi }}_0=0\hfill \end{array}$$

(61)

and the Noether conserved quantity (59) reduces to

$$\begin{array}{cc}\hfill I_L=& \left[ln\left(R_{\nu }{}_a{}^{C}D_{\Delta ,t}^{\alpha }{a}_{\nu }-B\right)-{\displaystyle \frac{\alpha R_{\nu }{}_a{}^{C}D_t^{\alpha }{a}_{\nu }}{R_{\nu }{}_a{}^{C}D_{\tau }^{\alpha }{a}_{\nu }-B}}\right]{\xi }_0\hfill \\ \hfill & +{\int }_a^t{\displaystyle \frac{1}{R_{\nu }{}_a{}^{C}D_{\tau }^{\alpha }{a}_{\nu }-B}}\left({\displaystyle \frac{\partial R_{\nu }}{\partial \tau }}{}_a{}^{C}D_{\tau }^{\alpha }{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial \tau }}\right){\xi }_0\mathrm{d}\tau \hfill \\ \hfill & -{\int }_a^t{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left[ln\left(R_{\nu }{}_a{}^{C}D_{\tau }^{\alpha }{a}_{\nu }-B\right)-{\displaystyle \frac{\alpha R_{\nu }{}_a{}^{C}D_{\tau }^{\alpha }{a}_{\nu }}{R_{\nu }{}_a{}^{C}D_{\tau }^{\alpha }{a}_{\nu }-B}}\right]{\xi }_0\mathrm{d}\tau \hfill \\ \hfill & -{\int }_a^t\left[{}_{\tau }D_b^{\alpha }\left({\displaystyle \frac{R_{\nu }}{R_{\nu }{}_a{}^{C}D_{\tau }^{\alpha }{a}_{\nu }-B}}\right){\xi }_{\omega }-{\displaystyle \frac{R_{\nu }{}_a{}^{C}D_{\tau }^{\alpha }{\xi }_{\nu }}{R_{\nu }{}_a{}^{C}D_{\tau }^{\alpha }{a}_{\nu }-B}}\right]\mathrm{d}\tau .\hfill \end{array}$$

(62)

If $\alpha =1$, Noether identity (56) reduces to

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{R_{\nu }{a}_{\nu }^{\Delta }-B}}\left[\left({\displaystyle \frac{\partial R_{\nu }}{\partial t}}{\xi }_0+{\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }^{\sigma }}}{\xi }_{\omega }^{\sigma }\right)a_{\nu }^{\Delta }+R_{\nu }{\xi }_{\nu }^{\Delta }\right.\left.-R_{\nu }{a}_{\nu }^{\Delta }{\xi }_0^{\Delta }-{\displaystyle \frac{\partial B}{\partial t}}{\xi }_0-{\displaystyle \frac{\partial B}{\partial a_{\omega }^{\sigma }}}{\xi }_{\omega }^{\sigma }\right]\hfill \\ \hfill & +ln\left(R_{\nu }{a}_{\nu }^{\Delta }-B\right){\xi }_0^{\Delta }=0\hfill \end{array}$$

(63)

and the Noether conserved quantity (59) reduces to

$$\begin{array}{cc}\hfill I_L=& \left\{ln\left(R_{\nu }{a}_{\nu }^{\Delta }-B\right)\right.\left.-{\displaystyle \frac{1}{R_{\nu }{a}_{\nu }^{\Delta }-B}}\left[R_{\nu }{a}_{\nu }^{\Delta }+\mu \left({\displaystyle \frac{\partial R_{\nu }}{\partial t}}{a}_{\nu }^{\Delta }-{\displaystyle \frac{\partial B}{\partial t}}\right)\right]\right\}{\xi }_0\hfill \\ \hfill & +{\int }_a^t{\displaystyle \frac{1}{R_{\nu }{a}_{\nu }^{\Delta }-B}}\left({\displaystyle \frac{\partial R_{\nu }}{\partial \tau }}{a}_{\nu }^{\Delta }-{\displaystyle \frac{\partial B}{\partial \tau }}\right){\xi }_0^{\sigma }\Delta \tau \hfill \\ \hfill & -{\int }_a^t\left\{ln\left(R_{\nu }{a}_{\nu }^{\Delta }-B\right)\right.{\left.-{\displaystyle \frac{1}{\left(R_{\nu }{a}_{\nu }^{\Delta }-B\right)}}\left[R_{\nu }{a}_{\nu }^{\Delta }+\mu \left({\displaystyle \frac{\partial R_{\nu }}{\partial \tau }}{a}_{\nu }^{\Delta }-{\displaystyle \frac{\partial B}{\partial \tau }}\right)\right]\right\}}^{\Delta }{\xi }_0^{\sigma }\Delta \tau \hfill \\ \hfill & +{\int }_a^t\left[{\displaystyle \frac{\Delta }{\Delta t}}\left({\displaystyle \frac{R_{\nu }}{R_{\nu }{a}_{\nu }^{\Delta }-B}}\right){\xi }_{\omega }^{\sigma }+{\displaystyle \frac{1}{R_{\nu }{a}_{\nu }^{\Delta }-B}}{R}_{\nu }{\xi }_{\nu }^{\Delta }\right]\Delta \tau .\hfill \end{array}$$

(64)

4.3. Power Birkhoffian System

Definition 6.

For any $\left[t_a,t_b\right]\subseteq \left[a,b\right]$, if

$$\begin{array}{cc}\hfill & {\int }_{t_a}^{t_b}\left[{\left(R_{\nu }\left(t,a_{\omega }^{\sigma }\right)C_a{a}_{\nu }-B\left(t,a_{\omega }^{\sigma }\right)\right)}^{1+\gamma }\right]\Delta t\hfill \\ \hfill =& {\int }_{\rho \left(t_a\right)}^{\rho \left(t_b\right)}\left[{\left(R_{\nu }\left(\overline{t},{\overline{a}}_{\omega }^{\overline{\sigma }}\left(\overline{t}\right)\right){\overline{C}}_a{\overline{a}}_{\nu }\left(\overline{t}\right)-B\left(\overline{t},{\overline{a}}_{\omega }^{\overline{\sigma }}\right)\right)}^{1+\gamma }\right]\overline{\Delta }\overline{t},\hfill \end{array}$$

(65)

then it is claimed that the action Equation (33) is invariant under the general infinitesimal transformation.

If Equation (33) is invariant under the transformation, then the generators ${\xi }_0$, ${\xi }_{\omega }$ satisfy the Noether identity

$$\begin{array}{cc}\hfill & \left(1+\gamma \right){\left(R_{\nu }{C}_a{a}_{\nu }-B\right)}^{\gamma }\left[\left({\displaystyle \frac{\partial R_{\nu }}{\partial t}}{\xi }_0+{\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }^{\sigma }}}{\xi }_{\omega }^{\sigma }\right)C_a{a}_{\nu }+R_{\nu }{C}_a{\xi }_{\nu }\right.\left.-\alpha R_{\nu }{C}_a{a}_{\nu }{\xi }_0^{\Delta }-{\displaystyle \frac{\partial B}{\partial t}}{\xi }_0-{\displaystyle \frac{\partial B}{\partial a_{\omega }^{\sigma }}}{\xi }_{\omega }^{\sigma }\right]\hfill \\ \hfill & +{\left(R_{\nu }{C}_a{a}_{\nu }-B\right)}^{1+\gamma }{\xi }_0^{\Delta }=0.\hfill \end{array}$$

(66)

Proof. 

Expanding Equation (65), we have

$$\begin{array}{cc}\hfill & {\int }_{t_a}^{t_b}\left[{\left(R_{\nu }\left(t,a_{\omega }^{\sigma }\right)C_a{a}_{\nu }-B\left(t,a_{\omega }^{\sigma }\right)\right)}^{1+\gamma }\right]\Delta t\hfill \\ \hfill =& {\int }_{\rho \left(t_a\right)}^{\rho \left(t_b\right)}\left[{\left(R_{\nu }\left(\overline{t},{\overline{a}}_{\omega }^{\overline{\sigma }}\left(\overline{t}\right)\right){\overline{C}}_a{\overline{a}}_{\nu }\left(\overline{t}\right)-B\left(\overline{t},{\overline{a}}_{\omega }^{\overline{\sigma }}\right)\right)}^{1+\gamma }\right]\overline{\Delta }\overline{t}\hfill \\ \hfill =& {\int }_{t_a}^{t_b}\left[{\left(R_{\nu }\left(T_{\epsilon },{\left(A_{\epsilon }^{\omega }\right)}^{\sigma }\right){\displaystyle \frac{C_a{A}_{\epsilon }^{\omega }}{{\left(T_{\epsilon }^{\Delta }\right)}^{\alpha }}}-B\left(T_{\epsilon },{\left(A_{\epsilon }^{\omega }\right)}^{\sigma }\right)\right)}^{1+\gamma }\right]T_{\epsilon }^{\Delta }\Delta t.\hfill \end{array}$$

(67)

Since $\left[t_a,t_b\right]\subseteq \left[a,b\right]$, Equation (67) is equal to

$$\begin{array}{cc}\hfill & {log}_b\left(R_{\nu }\left(t,a_{\omega }^{\sigma }\right)C_a{a}_{\nu }\left(t\right)-B\left(t,a_{\omega }^{\sigma }\right)\right)\hfill \\ \hfill =& {\left(R_{\nu }\left(T_{\epsilon },{\left(A_{\epsilon }^{\omega }\right)}^{\sigma }\right){\displaystyle \frac{C_a{A}_{\epsilon }^{\omega }}{{\left(T_{\epsilon }^{\Delta }\right)}^{\alpha }}}-B\left(T_{\epsilon },{\left(A_{\epsilon }^{\omega }\right)}^{\sigma }\right)\right)}^{1+\gamma }{T}_{\epsilon }^{\Delta }.\hfill \end{array}$$

(68)

Taking the derivative of Equation (68) with respect to $\epsilon$ and put $\epsilon =0$, then we get Equation (66). □

Theorem 3.

If Equation (33) is invariant under Definition 6, then

$$\begin{array}{cc}\hfill I_P=& \left\{{\left(\Im \right)}^{1+\gamma }-\left(1+\gamma \right){\left(\Im \right)}^{\gamma }\left[\alpha R_{\nu }{C}_a{a}_{\nu }\right.\right.\left.\left.+\mu \left({\displaystyle \frac{\partial R_{\nu }}{\partial t}}{C}_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial t}}\right)\right]\right\}{\xi }_0\hfill \\ \hfill & +{\int }_a^t\left(1+\gamma \right){\left(\Im \right)}^{\gamma }\left({\displaystyle \frac{\partial R_{\nu }}{\partial \tau }}{C}_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial \tau }}\right){\xi }_0^{\sigma }\Delta \tau \hfill \\ \hfill & -{\int }_a^t\left\{{\left(\Im \right)}^{1+\gamma }-\left(1+\gamma \right){\left(\Im \right)}^{\gamma }\left[\alpha R_{\nu }{C}_a{a}_{\nu }\right.\right.{\left.\left.+\mu \left({\displaystyle \frac{\partial R_{\nu }}{\partial t}}{C}_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial \tau }}\right)\right]\right\}}^{\Delta }{\xi }_0^{\sigma }\Delta \tau \hfill \\ \hfill & -{\int }_a^t\left\{{\Delta }_b\left[\left(1+\gamma \right){\left(\Im \right)}^{\gamma }{R}_{\nu }\right]{\xi }_{\omega }^{\sigma }-\left(1+\gamma \right){\left(\Im \right)}^{\gamma }{R}_{\nu }{C}_a{\xi }_{\nu }\right\}\Delta \tau \hfill \end{array}$$

(69)

is a Noether conserved quantity.

Proof. 

Taking the delta derivative of Equation (69), making use of Equations (40) and (66), we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{\Delta }{\Delta t}}{I}_P=& \left\{{\left(\Im \right)}^{1+\gamma }-\left(1+\gamma \right){\left(\Im \right)}^{\gamma }\left[\alpha R_{\nu }{C}_a{a}_{\nu }\right.\right.{\left.\left.+\mu \left({\displaystyle \frac{\partial R_{\nu }}{\partial t}}{C}_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial t}}\right)\right]\right\}}^{\Delta }{\xi }_0^{\sigma }\hfill \\ \hfill & +\left\{{\left(\Im \right)}^{1+\gamma }-\left(1+\gamma \right){\left(\Im \right)}^{\gamma }\left[\alpha R_{\nu }{C}_a{a}_{\nu }\right.\right.\left.\left.+\mu \left({\displaystyle \frac{\partial R_{\nu }}{\partial t}}{C}_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial t}}\right)\right]\right\}{\xi }_0^{\Delta }\hfill \\ \hfill & +\left(1+\gamma \right){\left(\Im \right)}^{\gamma }\left({\displaystyle \frac{\partial R_{\nu }}{\partial \tau }}{C}_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial \tau }}\right){\xi }_0^{\sigma }\hfill \\ \hfill & -\left\{{\left(\Im \right)}^{1+\gamma }-\left(1+\gamma \right){\left(\Im \right)}^{\gamma }\left[\alpha R_{\nu }{C}_a{a}_{\nu }\right.\right.{\left.\left.+\mu \left({\displaystyle \frac{\partial R_{\nu }}{\partial t}}{C}_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial \tau }}\right)\right]\right\}}^{\Delta }{\xi }_0^{\sigma }\hfill \\ \hfill & -{}_{\tau }D_{\Delta ,b}^{\alpha }\left[\left(1+\gamma \right){\left(\Im \right)}^{\gamma }{R}_{\nu }\right]{\xi }_{\omega }^{\sigma }+\left(1+\gamma \right){\left(\Im \right)}^{\gamma }{R}_{\nu }{C}_a{\xi }_{\nu }\hfill \\ \hfill =& \left[{\left(\Im \right)}^{\gamma }\left({\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }^{\sigma }}}·C_a{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial a_{\omega }^{\sigma }}}\right)+{\Delta }_b\left({\left(\Im \right)}^{\gamma }{R}_{\nu }\right)\right]{\xi }_k^{\sigma }\hfill \\ \hfill =& 0.\hfill \end{array}$$

(70)

Hence, Equation (69) is a Noether conserved quantity of the power Birkhoffian system. □

Remark 7.

If $\mathbb{T}=\mathbb{R}$, Noether identity (66) reduces to

$$\begin{array}{cc}\hfill & \left(1+\gamma \right){\left(R_{\nu }{}_a{}^{C}D_t^{\alpha }{a}_{\nu }-B\right)}^{\gamma }\left[\left({\displaystyle \frac{\partial R_{\nu }}{\partial t}}{\xi }_0+{\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }}}{\xi }_{\omega }\right){}_a{}^{C}D_t^{\alpha }{a}_{\nu }\right.\hfill \\ \hfill & \left.+R_{\nu }{}_a{}^{C}D_t^{\alpha }{\xi }_{\nu }-\alpha R_{\nu }{}_a{}^{C}D_t^{\alpha }{a}_{\nu }{\dot{\xi }}_0-{\displaystyle \frac{\partial B}{\partial t}}{\xi }_0-{\displaystyle \frac{\partial B}{\partial a_{\omega }}}{\xi }_{\omega }\right]\hfill \\ \hfill & +{\left(R_{\nu }{}_a{}^{C}D_t^{\alpha }{a}_{\nu }-B\right)}^{1+\gamma }{\dot{\xi }}_0=0\hfill \end{array}$$

(71)

and the Noether conserved quantity (69) reduces to

$$\begin{array}{cc}\hfill I_P=& \left[{\left(R_{\nu }{}_a{}^{C}D_t^{\alpha }{a}_{\nu }-B\right)}^{1+\gamma }\right.\left.-\left(1+\gamma \right){\left(R_{\nu }{}_a{}^{C}D_t^{\alpha }{a}_{\nu }-B\right)}^{\gamma }\left(\alpha R_{\nu }{}_a{}^{C}D_t^{\alpha }{a}_{\nu }\right)\right]{\xi }_0\hfill \\ \hfill & +{\int }_a^t\left(1+\gamma \right){\left(R_{\nu }{}_a{}^{C}D_{\tau }^{\alpha }{a}_{\nu }-B\right)}^{\gamma }\left({\displaystyle \frac{\partial R_{\nu }}{\partial \tau }}{}_a{}^{C}D_{\tau }^{\alpha }{a}_{\nu }-{\displaystyle \frac{\partial B}{\partial \tau }}\right){\xi }_0\mathrm{d}\tau \hfill \\ \hfill & -{\int }_a^t{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left\{{\left(R_{\nu }{}_a{}^{C}D_{\tau }^{\alpha }{a}_{\nu }-B\right)}^{1+\gamma }\right.\left.-\left(1+\gamma \right){\left(R_{\nu }{}_a{}^{C}D_{\tau }^{\alpha }{a}_{\nu }-B\right)}^{\gamma }\left(\alpha R_{\nu }{}_a{}^{C}D_t^{\alpha }{a}_{\nu }\right)\right\}{\xi }_0\mathrm{d}\tau \hfill \\ \hfill & -{\int }_a^t\left\{{}_{\tau }D_b^{\alpha }\left[\left(1+\gamma \right){\left(R_{\nu }{}_a{}^{C}D_t^{\alpha }{a}_{\nu }-B\right)}^{\gamma }{R}_{\nu }\right]{\xi }_{\omega }\right.\left.-\left(1+\gamma \right){\left(R_{\nu }{}_a{}^{C}D_{\tau }^{\alpha }{a}_{\nu }-B\right)}^{\gamma }{R}_{\nu }{}_a{}^{C}D_{\tau }^{\alpha }{\xi }_{\nu }\right\}\mathrm{d}\tau .\hfill \end{array}$$

(72)

If $\alpha =1$, Noether identity (66) reduces to

$$\begin{array}{cc}\hfill & \left(1+\gamma \right){\left(R_{\nu }{a}_{\nu }^{\Delta }-B\right)}^{\gamma }\left[\left({\displaystyle \frac{\partial R_{\nu }}{\partial t}}{\xi }_0+{\displaystyle \frac{\partial R_{\nu }}{\partial a_{\omega }^{\sigma }}}{\xi }_{\omega }^{\sigma }\right)a_{\nu }^{\Delta }+R_{\nu }{\xi }_{\nu }^{\Delta }\right.\hfill \\ \hfill & \left.-R_{\nu }{a}_{\nu }^{\Delta }{\xi }_0^{\Delta }-{\displaystyle \frac{\partial B}{\partial t}}{\xi }_0-{\displaystyle \frac{\partial B}{\partial a_{\omega }^{\sigma }}}{\xi }_{\omega }^{\sigma }\right]+{\left(R_{\nu }{a}_{\nu }^{\Delta }-B\right)}^{1+\gamma }{\xi }_0^{\Delta }=0\hfill \end{array}$$

(73)

and the Noether conserved quantity (69) reduces to

$$\begin{array}{cc}\hfill I_P=& \left\{{\left(R_{\nu }{a}_{\nu }^{\Delta }-B\right)}^{1+\gamma }-\left(1+\gamma \right){\left(R_{\nu }{a}_{\nu }^{\Delta }-B\right)}^{\gamma }\right.\left.\left[R_{\nu }{a}_{\nu }^{\Delta }+\mu \left({\displaystyle \frac{\partial R_{\nu }}{\partial t}}{a}_{\nu }^{\Delta }-{\displaystyle \frac{\partial B}{\partial t}}\right)\right]\right\}{\xi }_0\hfill \\ \hfill & +{\int }_a^t\left(1+\gamma \right){\left(R_{\nu }{a}_{\nu }^{\Delta }-B\right)}^{\gamma }\left({\displaystyle \frac{\partial R_{\nu }}{\partial \tau }}{a}_{\nu }^{\Delta }-{\displaystyle \frac{\partial B}{\partial \tau }}\right){\xi }_0^{\sigma }\Delta \tau \hfill \\ \hfill & -{\int }_a^t\left\{{\left(R_{\nu }{a}_{\nu }^{\Delta }-B\right)}^{1+\gamma }-\left(1+\gamma \right){\left(R_{\nu }{a}_{\nu }^{\Delta }-B\right)}^{\gamma }\right.{\left.\left[R_{\nu }{a}_{\nu }^{\Delta }+\mu \left({\displaystyle \frac{\partial R_{\nu }}{\partial t}}{a}_{\nu }^{\Delta }-{\displaystyle \frac{\partial B}{\partial \tau }}\right)\right]\right\}}^{\Delta }{\xi }_0^{\sigma }\Delta \tau \hfill \\ \hfill & +{\int }_a^t\left\{{\displaystyle \frac{\Delta }{\Delta t}}\left[\left(1+\gamma \right){\left(R_{\nu }{a}_{\nu }^{\Delta }-B\right)}^{\gamma }{R}_{\nu }\right]{\xi }_{\omega }^{\sigma }\right.\left.+\left(1+\gamma \right){\left(R_{\nu }{a}_{\nu }^{\Delta }-B\right)}^{\gamma }{R}_{\nu }{\xi }_{\nu }^{\Delta }\right\}\Delta \tau .\hfill \end{array}$$

(74)

5. Examples

Considering the fractional Hojman–Urrutia model [34,35] on time scales, where $\mathbb{T}=\left\{a+mh:m\in \mathbb{N}\right\}$, the Birkhoffian and Birkhoffian functions are

$$\begin{array}{cc}\hfill & B={\displaystyle \frac{1}{2}}\left[{\left(a_2^{\sigma }\right)}^2+2a_2^{\sigma }{a}_3^{\sigma }-{\left(a_4^{\sigma }\right)}^{\sigma }\right],\hfill \\ \hfill & R_1=a_2^{\sigma }+a_3^{\sigma },R_2=0,R_3=a_4^{\sigma },R_4=0.\hfill \end{array}$$

(75)

Next, we investigate Noether symmetries under exponential, logarithmic, and power Birkhoffian separately.

For exponential Birkhoffian, the Pfaff action is

$$A_E={\int }_a^b\left[exp\left(R_{\nu }\left(t,a_{\omega }^{\sigma }\right)C_a{a}_{\nu }-B\left(t,a_{\omega }^{\sigma }\right)\right)\right]\Delta t.$$

(76)

Equation (18) gives the equation of motion:

$$\begin{array}{cc}\hfill & {\Delta }_b\left[exp\left(R_1{C}_a{a}_1-B\right)R_1\right]=0,\hfill \\ \hfill & exp\left(R_1{C}_a{a}_1-B\right)\left(C_a{a}_1-a_2^{\sigma }-a_3^{\sigma }\right)=0,\hfill \\ \hfill & exp\left(R_1{C}_a{a}_1-B\right)\left(C_a{a}_1-a_2^{\sigma }\right)+{\Delta }_b\left[exp\left(a_4^{\sigma }{C}_a{a}_3-B\right)a_4^{\sigma }\right],\hfill \\ \hfill & exp\left(a_4^{\sigma }{C}_a{a}_3-B\right)\left(C_a{a}_3+a_4^{\sigma }\right)=0.\hfill \end{array}$$

(77)

The Noether identity Equation (46) gives

$$\begin{array}{cc}\hfill & exp\left(R_1{C}_a{a}_1-B\right)\left({\xi }_0^{\Delta }+{\xi }_2^{\sigma }{C}_a{a}_1+{\xi }_3^{\sigma }{C}_a{a}_1\right.\left.+R_1{C}_a{\xi }_1-\alpha R_1{C}_a{a}_1{\xi }_0^{\Delta }-R_1{\xi }_2^{\sigma }-a_2^{\sigma }{\xi }_3^{\sigma }\right)\hfill \\ \hfill & +exp\left(R_3{C}_a{a}_3-B\right)\left({\xi }_0^{\Delta }+{\xi }_4^{\sigma }{C}_a{a}_3\right.\left.+a_4^{\sigma }{C}_a{\xi }_3-\alpha a_4^{\sigma }{C}_a{a}_3{\xi }_0^{\Delta }+a_4^{\sigma }{\xi }_4^{\sigma }\right).\hfill \end{array}$$

(78)

Equation (78) can obtain the following solution

$${\xi }_0=1,{\xi }_1=1,{\xi }_2=1,{\xi }_3=0,{\xi }_4=1.$$

(79)

Theorem 1 gives the following conserved quantity

$$\begin{array}{cc}\hfill I_E=& E_m-\alpha \left[\left(a_2^{\sigma }+a_3^{\sigma }\right)C_a{a}_1+a_4^{\sigma }{C}_a{a}_3\right]\hfill \\ \hfill & +{\int }_a^t{\left\{E_m^{\Delta }-\alpha \left[\left(a_2^{\sigma }+a_3^{\sigma }\right)C_a{a}_1+a_4^{\sigma }{C}_a{a}_3\right]\right.}^{\Delta }\left.+{\mathbb{D}}_{\Delta }^{\alpha }\left[1,E_1\left(a_2^{\sigma }+a_3^{\sigma }\right)\right]\right\}\Delta \tau \hfill \\ \hfill =& \mathrm{const}.\hfill \end{array}$$

(80)

where $E_x=exp\left(R_x{C}_a{a}_x-B\right),E_m=\sum E_x$, ${\mathbb{D}}_{\Delta }^{\alpha }\left[f,g\right]=g{C}_{a}f-f^{\sigma }{\Delta }_{b}g$.

For logarithmic Birkhoffian, the Pfaff action is

$$A_L={\int }_a^b\left[lo{g}_b\left(R_{\nu }\left(t,a_{\omega }^{\sigma }\right)C_a{a}_{\nu }-B\left(t,a_{\omega }^{\sigma }\right)\right)\right]\Delta t.$$

(81)

Equation (29) gives the equation of motion:

$$\begin{array}{cc}\hfill & {\Delta }_b\left({\displaystyle \frac{R_1}{R_1{C}_a{a}_1-B}}\right)=0,\hfill \\ \hfill & {\displaystyle \frac{1}{R_1{C}_a{a}_1-B}}\left(C_a{a}_1-a_2^{\sigma }-a_3^{\sigma }\right)=0,\hfill \\ \hfill & {\displaystyle \frac{1}{R_1{C}_a{a}_1-B}}\left(C_a{a}_1-a_2^{\sigma }\right)+{\Delta }_b\left({\displaystyle \frac{a_4^{\sigma }}{a_4^{\sigma }{C}_a{a}_3-B}}\right),\hfill \\ \hfill & {\displaystyle \frac{1}{a_4^{\sigma }{C}_a{a}_3-B}}\left(C_a{a}_3+a_4^{\sigma }\right)=0.\hfill \end{array}$$

(82)

The Noether identity Equation (56) gives

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{R_1{C}_a{a}_1-B}}\left({\xi }_2^{\sigma }{C}_a{a}_1+{\xi }_3^{\sigma }{C}_a{a}_1\right.\left.+R_1{C}_a{\xi }_1-\alpha R_1{C}_a{a}_1{\xi }_0^{\Delta }-R_1{\xi }_2^{\sigma }-a_2^{\sigma }{\xi }_3^{\sigma }\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{R_3{C}_a{a}_3-B}}\left({\xi }_4^{\sigma }{C}_a{a}_3\right.\left.+a_4^{\sigma }{C}_a{\xi }_3-\alpha a_4^{\sigma }{C}_a{a}_3{\xi }_0^{\Delta }+a_4^{\sigma }{\xi }_4^{\sigma }\right)\hfill \\ \hfill & +ln\left(R_1{C}_a{a}_1-B\right)\left(R_3{C}_a{a}_3-B\right){\xi }_0^{\Delta }=0.\hfill \end{array}$$

(83)

Equation (83) can also obtain the solution (79). Theorem 2 gives the following conserved quantity

$$\begin{array}{cc}\hfill I_L=& ln\left({\Im }_m\right)-\alpha \left[\left(a_2^{\sigma }+a_3^{\sigma }\right)C_a{a}_1+a_4^{\sigma }{C}_a{a}_3\right]\hfill \\ \hfill & +{\int }_a^t{\left\{{ln}^{\Delta }\left({\Im }_m\right)-\alpha \left[\left(a_2^{\sigma }+a_3^{\sigma }\right)C_a{a}_1+a_4^{\sigma }{C}_a{a}_3\right]\right.}^{\Delta }\left.+{\mathbb{D}}_{\Delta }^{\alpha }\left[1,{\Im }_1^{-1}\left(a_2^{\sigma }+a_3^{\sigma }\right)\right]\right\}\Delta \tau \hfill \\ \hfill =& \mathrm{const}.\hfill \end{array}$$

(84)

where ${\Im }_x=R_x{C}_a{a}_x-B,{\Im }_m=\Pi {\Im }_x$.

For power Birkhoffian, the Pfaff action is

$$A_P={\int }_a^b\left[{\left(R_{\nu }\left(t,a_{\omega }^{\sigma }\right)C_a{a}_{\nu }-B\left(t,a_{\omega }^{\sigma }\right)\right)}^{1+\gamma }\right]\Delta t.$$

(85)

Equation (40) gives the equation of motion:

$$\begin{array}{cc}\hfill & {\Delta }_b\left[{\left(R_1{C}_a{a}_1-B\right)}^{\gamma }{R}_1\right]=0,\hfill \\ \hfill & {\left(R_1{C}_a{a}_1-B\right)}^{\gamma }\left(C_a{a}_1-a_2^{\sigma }-a_3^{\sigma }\right)=0,\hfill \\ \hfill & {\left(R_1{C}_a{a}_1-B\right)}^{\gamma }\left(C_a{a}_1-a_2^{\sigma }\right)+{\Delta }_b\left[{\left(a_4^{\sigma }{C}_a{a}_3-B\right)}^{\gamma }{a}_4^{\sigma }\right],\hfill \\ \hfill & {\left(a_4^{\sigma }{C}_a{a}_3-B\right)}^{\gamma }\left(C_a{a}_3+a_4^{\sigma }\right)=0.\hfill \end{array}$$

(86)

The Noether identity Equation (66) gives

$$\begin{array}{cc}\hfill & \left(1+\gamma \right){\left(R_1{C}_a{a}_1-B\right)}^{\gamma }\left({\xi }_2^{\sigma }{C}_a{a}_1+{\xi }_3^{\sigma }{C}_a{a}_1\right.\left.+R_1{C}_a{\xi }_1-\alpha R_1{C}_a{a}_1{\xi }_0^{\Delta }-R_1{\xi }_2^{\sigma }-a_2^{\sigma }{\xi }_3^{\sigma }\right)\hfill \\ \hfill & \left(1+\gamma \right){\left(R_1{C}_a{a}_1-B\right)}^{\gamma }\left({\xi }_4^{\sigma }{C}_a{a}_3\right.\left.+a_4^{\sigma }{C}_a{\xi }_3-\alpha a_4^{\sigma }{C}_a{a}_3{\xi }_0^{\Delta }+a_4^{\sigma }{\xi }_4^{\sigma }\right)\hfill \\ \hfill & +{\left(R_1{C}_a{a}_1-B\right)}^{1+\gamma }\left(R_3{C}_a{a}_3-B\right){\xi }_0^{\Delta }=0.\hfill \end{array}$$

(87)

Equation (87) can also obtain the solution (79). By Theorem 3, we have the following Noether conserved quantity

$$\begin{array}{cc}\hfill I_P=& -\left(1+\gamma \right){\Im }_1^{\gamma }\alpha \left[\left(a_2^{\sigma }+a_3^{\sigma }\right)C_a{a}_1+a_4^{\sigma }{C}_a{a}_3\right]\hfill \\ \hfill & +{\int }_a^t\left\{{\left\{\left(1+\gamma \right){\Im }_1^{\gamma }\alpha \left[\left(a_2^{\sigma }+a_3^{\sigma }\right)C_a{a}_1+a_4^{\sigma }{C}_a{a}_3\right]\right\}}^{\Delta }\right.\left.+{\mathbb{D}}_{\Delta }^{\alpha }\left[1,\left(1+\gamma \right){\Im }_1^{\gamma }\left(a_2^{\sigma }+a_3^{\sigma }\right)\right]\right\}\Delta \tau \hfill \\ \hfill =& \mathrm{const}.\hfill \end{array}$$

(88)

If $a=0$, $b=11$, $\alpha =0.5$ and $h=1$ and $t\in \left[1,10\right]$. Given initial conditions $a_1\left(1\right)=1$, $a_2\left(1\right)=1$, $a_3\left(1\right)=1$, $a_4\left(1\right)=1$, then we can get the following simulation Figure 1.

The results indirectly prove the findings in reference [21].

6. Conclusions

The application of fractional theory and time-scale theory to the study of dynamical problems has resulted in the establishment of a more widely appropriate fractional time-scales model. Noether’s theorem for non-standard Birkhoffian system under Caputo $\Delta$ derivatives is discussed in the paper and several special cases are given; some of the special case results are consistent with the original results and some are new. Given the unifying and extensibility features exhibited by time-scale theory, the methods and results presented in this paper have the potential for a much wider range of applications that can be extended to different complex systems. Non-standard functions can be used to model non-linear, non-conservative dynamical systems. However, it is clear that further refinement is required in subsequent research, for example in terms of the specific physical meaning of conserved quantities in practical problems.

The investigation of the fractional time-scales theory is still in its infancy, with a considerable amount of work yet to be completed. For instance, there are the Lie symmetry and Hojman conserved quantities, the Mei symmetry and Mei conserved quantities on fractional time scales, methods for extending the theory to non-holonomic systems, and so forth.